You have a mortgage loan for 65% of the $2.5 million purchase price of an apartment. The monthly payment will be $10,400 and the mortgage term is 30 years. What is the EAR on this loan?单项选择题
A
A. 6.82%
B
B. 6.25%
C
C. 6.46%
D
D. 6.91%

Question

You have a mortgage loan for 65% of the $2.5 million purchase price of an apartment. The monthly payment will be $10,400 and the mortgage term is 30 years. What is the EAR on this loan?

BlackTom AI · Accepted Answer

Use the present-value-of-an-annuity formula to find the monthly interest rate, then convert to EAR. Relevant concepts: PV of level monthly payments for n months is PV = PMT * [1 - (1 + r)^(-n)] / r. EAR (effective annual rate) = (1 + r)^12 - 1 where r is the monthly rate.

Step-by-step reasoning and calculations:
1. Determine loan amount and parameters:
- Purchase price = $2,500,000
- Loan = 65% × $2,500,000 = $1,625,000 (PV)
- Monthly payment PMT = $10,400
- Term = 30 years = 360 months (n)

2. Compute the annuity factor needed:
PV / PMT = 1,625,000 / 10,400 = 156.25
So we need r satisfying [1 - (1 + r)^(-360)] / r = 156.25.

3. Solve for monthly rate r by trial / interpolation:
Evaluate at r = 0.0055 (0.55% monthly):
(1 + r)^360 ≈ e^{360·ln(1.0055)} ≈ 7.20 → (1 + r)^(-360) ≈ 0.1389
Annuity factor ≈ (1 - 0.1389)/0.0055 ≈ 0.8611/0.0055 = 156.56

At r = 0.0056 (0.56% monthly):
(1 + r)^360 ≈ 7.468 → (1 + r)^(-360) ≈ 0.1339
Annuity factor ≈ (1 - 0.1339)/0.0056 ≈ 0.8661/0.0056 = 154.66

The target 156.25 lies between these; interpolate:
r ≈ 0.0055 + (156.56 − 156.25) / (156.56 − 154.66) × 0.0001
r ≈ 0.0055 + (0.31 / 1.90) × 0.0001 ≈ 0.0055163 (monthly rate)

4. Convert monthly rate to EAR:
EAR = (1 + r)^12 − 1 = (1.0055163)^12 − 1 ≈ e^{12·ln(1.0055163)} − 1
ln(1.0055163) ≈ 0.0055011 → 12· = 0.066013 → e^{0.066013} ≈ 1.06824
EAR ≈ 0.06824 = 6.824% ≈ 6.82%.

Why answer A is correct and others incorrect:
- A (6.82%) matches the EAR computed from the exact annuity relation.
- B (6.25%), C (6.46%), D (6.91%) are inconsistent with the payment/loan/term combination; they either under- or overstate the monthly rate and thus produce annuity factors that do not equal 156.25.

Final confirmation: The correct EAR is 6.82% (Choice A).

You have a mortgage loan for 65% of the $2.5 million purchase price of an apartment. The monthly payment will be $10,400 and the mortgage term is 30 years. What is the EAR on this loan?单项选择题

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